Question

How can the zeros and end behavior of a polynomial function allow a graph to be sketched?

Answer 1

Given a polynomial of the following form:

$f(x)\text{ = (x-1)}(x+1)^2(x^2-4)$

Equaling f(x) to zero:

$0=(x-1)(x+1)^2(x^2-4)$

If we know that f(x) is equal to zero, then at least one term or all terms are equal to zero.

Equaling each term to zero as shown as follows:

x-1 = 0 (x+1)^2 = 0 x^2-4 = 0

Isolating the x:

x=1 x=-1 x=± 2

Now, looking at the degree of the power of each term.

(x-1)^1 --> m=1 [odd] (x+1)^2--> m=2 [even] (x^2-4)^1-->m=1 [odd]

Now, we can draw the graph: